This paper discusses the design of observer-based reduced order controllers for the stabilization of large scale linear discrete-time control systems. This design is carried out via deriving a reduced-order model for the given linear plant using the dominant state of the linear plant. Using this reduced-order linear model, sufficient conditions are derived for the design of observer-based reduced order controllers. A separation principle has been established in this paper which demonstrates that the observer poles and controller poles can be separated and hence the pole-placement problem and observer design are independent of each other.

1. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.1, No.5, December 2011
DOI : 10.5121/ijcseit.2011.1508 85
OBSERVER-BASED REDUCED ORDER CONTROLLER
DESIGN FOR THE STABILIZATION OF LARGE SCALE
LINEAR DISCRETE-TIME CONTROL SYSTEMS
Sundarapandian Vaidyanathan1
and Kavitha Madhavan2
1
Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University
Avadi, Chennai-600 062, Tamil Nadu, INDIA
sundarvtu@gmail.com
2
Department of Mathematics, Vel Tech Dr. RR & Dr. SR Technical University
Avadi, Chennai-600 062, Tamil Nadu, INDIA
kavi78_m@yahoo.com
ABSTRACT
This paper discusses the design of observer-based reduced order controllers for the stabilization of large
scale linear discrete-time control systems. This design is carried out via deriving a reduced-order model
for the given linear plant using the dominant state of the linear plant. Using this reduced-order linear
model, sufficient conditions are derived for the design of observer-based reduced order controllers. A
separation principle has been established in this paper which demonstrates that the observer poles and
controller poles can be separated and hence the pole-placement problem and observer design are
independent of each other.
KEYWORDS
Dominant State, Reduced-Order Model, Stabilization, Observers, Discrete-Time Linear System.
1. INTRODUCTION
The reduced order controller design and observer design are active research problems in the linear
systems literature and there has been a significant attention paid in the literature on these two
problems during the past four decades [1-10]. In the recent decades, there has been a
considerable attention paid to the control problem of large scale linear systems. This is mainly
owing to the practical and technical issues like data acquisition, sensing, computing facilities,
information transfer networks and the cost associated with them, which arise from the use of full
order controller design. Thus, there is a great demand in the industry for the control of large scale
linear systems with the use of reduced-order controllers rather than full-order controllers.
A recent approach for obtaining the reduced-order controllers is via the reduced-order model of a
linear plant preserving the dynamic as well as static properties of the system and then devising
controllers for the reduced-order model thus obtained [4-8]. This approach has practical and

2. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.1, No.5, December 2011
86
technical benefits for the reduced-order controller design for large-scale linear systems with high
dimension and complexity.
The observer-based reduced-order controller design is motivated by the fact that the dominant
state of the linear plant may not be available for measurement and hence for implementing the
pole placement law, only the reduced order exponential observer can be used in lieu of the
dominant state of the given discrete-time linear system.
In this paper, we derive a reduced-order model for any linear discrete-time control system and our
approach is based on the approach of using the dominant state of the given linear discrete-time
control system, i.e. we derive the reduced-order model for a given discrete-time linear control
system keeping only the dominant state of the given discrete-time linear control system. Using
the reduced-order model obtained, we characterize the existence of a reduced-order exponential
observer that tracks the state of the reduced-order model, i.e. the dominant state of the original
linear plant. We note that the reduced-order observer design detailed in this paper is a discrete-
time analogue of the results of Aldeen and Trinh [8] for the observer design of the dominant state
of continuous-time linear control systems.
Using the reduced-order model of the original linear plant, we also characterize the existence of a
stabilizing feedback control law that uses only the dominant state of the original plant. Also,
when the plant is stabilizable by a state feedback control law, the full information of the dominant
state may not be always available. For this reason, we establish a separation principle so that the
state of the exponential observer may be used in lieu of the dominant state of the original linear
plant, which facilitates the implementation of the stabilizing feedback control law derived. The
design of observer-based reduced order controllers for large scale discrete-time linear systems
derived in this paper has important practical, industrial applications.
This paper is organized as follows. In Section 2, we describe the construction of the reduced-
order plant of a given linear discrete-time control system. In Section 3, we derive necessary and
sufficient conditions for the exponential observer design for the reduced-order linear control
plant. In Section 4, we derive necessary and sufficient conditions for the reduced-order plant to be
stabilizable by a linear feedback control law and we also present a separation principle for the
observer-based reduced order controller design. In Section 5, we describe a numerical example.
In Section 6, we summarize the main results derived in this paper.
2. REDUCED ORDER MODEL FOR DISCRETE-TIME LINEAR SYSTEMS
In this section, we consider a large scale linear discrete-time control system 1S given by
( 1) ( ) ( )
( ) ( )
x k Ax k Bu k
y k Cx k
+ = +
=
(1)
where n
x R∈ is the state, m
u R∈ is the control or input and p
y ∈R is the system output. We
assume that ,A B and C are constant matrices with real entries having dimensions
,n n n m× × p n× respectively.
First, we suppose that we have made an identification of the dominant (slow) and non-
dominant (fast) states of the original linear system (1) using the modal approach as
described in [9].
Without loss of generality, we may assume that

3. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.1, No.5, December 2011
87
,
s
f
x
x
x
 
=  
 
where r
sx R∈ represents the dominant state and n r
fx R −
∈ represents the non-dominant
state of the system.
Then the linear system (1) becomes
( 1) ( )
( )
( 1) ( )
( )
( )
( )
s ss sf s s
f fs ff f f
s
s f
f
x k A A x k B
u k
x k A A x k B
x k
y k C C
x k
+       
= +       +       
 
 =   
 
(2)
From (2), we can write the plant equations as
( 1) ( ) ( ) ( )
( 1) ( ) ( ) ( )
( ) ( ) ( )
s ss s sf f s
f fs s ff f f
s s f f
x k A x k A x k B u k
x k A x k A x k B u k
y k C x k C x k
+ = + +
+ = + +
= +
(3)
Next, we shall assume that the matrix A has a set of n linearly independent eigenvectors.
In most practical situations, the matrix A has distinct eigenvalues and this condition is
immediately satisfied.
Thus, it follows that A is diagonalizable.
Thus, we can find a non-singular (modal) matrix P consisting of n linearly independent
eigenvectors of A such that
1
,P AP−
= Λ
where Λ is a diagonal matrix consisting of the n eigenvalues of .A
Now, we introduce a new set of coordinates on the state space given by
1
P xξ −
= (4)
In the new coordinates, the plant (1) becomes
1
( 1) ( ) ( )
( ) ( )
k k P Bu k
y k CP k
ξ ξ
ξ
−
+ = Λ +
=
Thus, we have

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88
1
( 1) 0 ( )
( )
( 1) 0 ( )
( )
( )
( )
s s s
f f f
s
f
k k
P Bu k
k k
k
y k CP
k
ξ ξ
ξ ξ
ξ
ξ
−
+ Λ     
= +     + Λ     
 
=  
 
(5)
where sΛ and fΛ are r r× and ( ) ( )n r n r− × − diagonal matrices respectively, consisting of the
dominant and non-dominant eigenvalues of .A
Define matrices , ,s f sΓ Γ Ψ and fΨ by
1 s
f
P B−
Γ 
=  Γ 
and s fCP  = Ψ Ψ  (6)
where , ,s f sΓ Γ Ψ and fΨ are , ( ) ,r m n r m p r× − × × and ( )p n r× − matrices respectively.
From (5) and (6), we see that the plant (3) has the following simple form in the new coordinates
(4).
( 1) ( ) ( )
( 1) ( ) ( )
( ) ( ) ( )
s s s s
f f f f
s s f f
k k u k
k k u k
y k k k
ξ ξ
ξ ξ
ξ ξ
+ = Λ + Γ
+ = Λ + Γ
= Ψ + Ψ
(7)
Next, we make the following assumptions:
(H1) As ,k → ∞ ( 1) ( ),f fk kξ ξ+ ≈ i.e. fξ takes a constant value in the steady-state.
(H2) The matrix fI − Λ is invertible.
Then it follows from (7) that for large values of ,k we have
( ) ( ) ( )f f f fk k u kξ ξ≈ Λ + Γ
i.e.
1
( ) ( ) ( )f f fk I u kξ −
≈ − Λ Γ (8)
Substituting (8) into (7), we obtain the reduced-order model of the linear plant (1) in the
ξ coordinates as
1
( 1) ( ) ( )
( ) ( ) ( ) ( )
s s s s
s s f f f
k k u k
y k k I u k
ξ ξ
ξ −
+ = Λ + Γ
= Ψ + Ψ − Λ Γ
(9)

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To obtain the reduced-order model of the linear plant (1) in the x coordinates, we proceed as
follows.
By the linear change of coordinates (4), it follows that
1
.P x Qxξ −
= =
Thus, we have
( ) ( ) ( )
( ) ( ) ( )
s s ss sf s
f f fs ff f
k x k Q Q x k
Q
k x k Q Q x k
ξ
ξ
       
= =       
       
(10)
Using (9) and (10), it follows that
1
( ) ( ) ( ) ( ) ( )f fs s ff f f fk Q k Q k I u kξ ξ ξ −
= + = − Λ Γ
or
1
( ) ( ) ( ) ( )ff f fs s f fQ k Q x k I u kξ −
= − + − Λ Γ (11)
Next, we assume the following:
(H3) The matrix ffQ is invertible.
Using the assumption (H3), the equation (11) becomes
1 1 1
( ) ( ) ( ) ( )f ff fs s ff f fx k Q Q x k Q I u k− − −
= − + − Λ Γ (12)
To simplify the notation, we define the matrices
1
ff fsR Q Q−
= − and 1 1
( )fff fS Q I− −
= − Λ Γ (13)
Using (13), the equation (12) can be simplified as
( ) ( ) ( )f sx k Rx k Su k= + (14)
Substituting (14) into (3), we obtain the reduced-order model 2S of the given linear system 1S as
( 1) ( ) ( )
( ) ( ) ( )
s s s s
s s s
x k A x k B u k
y k C x k D u k
∗ ∗
∗ ∗
+ = +
= +
(15)
where the matrices , ,s s sA B C∗ ∗ ∗
and sD∗
are defined by

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90
s ss sf
s s sf
s s f
s f
A A A R
B B A S
C C C R
D C S
∗
∗
∗
∗
= +
= +
= +
=
(16)
3. REDUCED ORDER OBSERVER DESIGN
In this section, we state a new result that prescribes a simple procedure for estimating the
dominant state of the given linear control system 1S that satisfies the assumptions (H1)-(H3)
described in Section 2.
Theorem 1. Let 1S be the linear system described by
( 1) ( ) ( )
( ) ( )
x k Ax k Bu k
y k Cx k
+ = +
=
(17)
Under the assumptions (H1)-(H3), the reduced-order model 2S of the linear system 1S can be
obtained (see Section 2) as
( 1) ( ) ( )
( ) ( ) ( )
s s s
s s
x k A x k B u k
y k C x k D u k
∗ ∗
∗ ∗
+ = +
= +
(18)
where , ,s s sA B C∗ ∗ ∗
and sD∗
are defined as in (16).
To estimate the dominant state sx of the system 1,S consider the candidate observer 3S defined
by
( 1) ( ) ( ) ( ) ( ) ( )s s s s s s s sz k A z k B u k K y k C z k D u k∗ ∗ ∗ ∗ ∗
 + = + + − −  (19)
Let the estimation error be defined as
s se z x= −
Then ( ) 0e k → exponentially as k → ∞ if and only if the matrix sK∗
is such that
s s sE A K C∗ ∗ ∗
= − is convergent. If ( ),s sC A∗ ∗
is observable, then we can always construct an
exponential observer of the form (19) having any desired speed of convergence.
Proof. From Eq. (2), we have
( 1) ( ) ( ) ( )s ss s sf f sx k A x k A x k B u k+ = + + (20)
Adding and subtracting the term ( ) ( )sf s f sA K C Rx k∗
− in the RHS of Eq. (20), we get

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( 1) ( ) ( ) ( ) ( ) ( ) ( )s ss sf s f s sf f sf s f s sx k A A R K C x k A x k A K C Rx k B u k∗ ∗
+ = + − + − − + (21)
Subtracting (21) from (19) and simplifying using the definitions (16), we get
( 1) ( ) ( ) ( )[ ( ) ( ) ( )]s s s sf s f f se k A K C e k A K C x k Rx k Su k∗ ∗ ∗ ∗
+ = − − − − − (22)
As proved in Section 2, the assumptions (H1)-(H3) yield
( ) ( ) ( )f sx k Rx k Su k≈ + (23)
Substituting (23) into (22), we get
( 1) ( ) ( ) ( )s s se k A K C e k Ee k∗ ∗ ∗
+ = − = (24)
which yields
( ) ( ) (0) (0)k k
s s se k A K C e E e∗ ∗ ∗
= − = (25)
From Eq. (25), it follows that ( ) 0e k → as k → ∞ if and only if E is convergent.
If ( ),s sC A∗ ∗
is observable, then it is well-known [10] that we can find an observer gain matrix sK∗
that will arbitrarily assign the eigenvalues of the error matrix s s sE A K C∗ ∗ ∗
= − . In particular, it
follows from Eq. (25) that we can find an exponential observer of the form (19) having any
desired speed of convergence.
4. OBSERVER-BASED REDUCED ORDER CONTROLLER DESIGN
In this section, we first state an important result that prescribes a simple procedure for stabilizing
the dominant state of the reduced-order linear plant derived in Section 2.
Theorem 2. Suppose that the assumptions (H1)-(H3) hold. Let 1S and 2S be defined as in
Theorem 1. For the reduced-order model 2 ,S the state feedback control law
( ) ( )s su k F x k∗
= − (26)
stabilizes the dominant state sx of the reduced-order model 2S having any desired speed of
convergence.
In practical applications, the dominant state sx of the reduced-order model 2S may not be directly
available for measurement and hence we cannot implement the state feedback control law (26).
To overcome this practical difficulty, we derive an important theorem, usually called as the
Separation Principle, which first establishes that the observer-based reduced-order controller
indeed stabilizes the dominant state of the given linear control system 1S and also demonstrates
that the observer poles and the closed-loop controller poles can be separated.

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Theorem 3 (Separation Principle). Suppose that the assumptions (H1)-(H3) hold. Suppose that
there exist matrices sF∗
and sK∗
such that s s sA B F∗ ∗ ∗
− and s s sA K C∗ ∗ ∗
− are both convergent
matrices. By Theorem 1, we know that the system 3S defined by (19) is an exponential observer
for the dominant state sx of the control system 1.S Then the observer poles and the closed-loop
controller poles are separated and the control law
( ) ( )s su k F z k∗
= − (27)
also stabilizes the dominant state sx of the control system 1.S
Proof. Under the feedback control law (27), the observer dynamics (19) of the system 3S
becomes
( )( 1) ( ) ( ) ( )s s s s s s s s s s s s s f fz k A B F K C K D F z k K C x k C x k∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
 + = − − − + +  (28)
By (14), we know that
( ) ( ) ( ) ( ) ( )f s s s sx k Rx k Su k Rx k SF z k∗
≈ + = − (29)
Substituting (29) into (28) and simplifying using the definitions (16), we get
( 1) ( ) ( ) ( )s s s s s s s s s sz k A B F K C z k K C x k∗ ∗ ∗ ∗ ∗ ∗ ∗
+ = − − + (30)
Substituting the control law (27) into (3), we also obtain
( 1) ( ) ( )s s s s s sx k A x k B F z k∗ ∗ ∗
+ = − (31)
In matrix representation, we can write equations (30) and (31) as
( 1) ( )
( 1) ( )
s ss s s
s ss s s s s s s
x k x kA B F
z k z kK C A B F K C
∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗
+  −   
=     + − −    
(32)
Since the estimation error e is defined by ,s se z x= − it is easy to see from equation (32) that the
error e satisfies the equation
( )( 1) ( )s s se k A K C e k∗ ∗ ∗
+ = − (33)
Using the ( , )x e coordinates, the composite system (32) can be simplified as
( 1) ( ) ( )
( 1) ( ) ( )0
s s ss s s s s
s s ss s s
x k x k x kA B F B F
M
e k e k e kA K C
∗ ∗ ∗ ∗ ∗
∗ ∗ ∗
+  − −     
= =      + −      
(34)
where

9. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.1, No.5, December 2011
93
.
0
s s s s s
s s s
A B F B F
M
A K C
∗ ∗ ∗ ∗ ∗
∗ ∗ ∗
 − −
=  
− 
(35)
Since the matrix M is block-triangular, it is immediate that
( ) ( )eig( ) eig eigs s s s s sM A B F A K C∗ ∗ ∗ ∗ ∗ ∗
= − −U (36)
which establishes the first part of the Separation Principle namely that the observer poles are
separated from the closed-loop controller poles.
To show that the observed-based control law (27) indeed works, we note that the closed-loop
regulator matrix s s sA B F∗ ∗ ∗
− and s s sA K C∗ ∗ ∗
− are both convergent matrices. From Eq. (36), it is
immediate that M is also a convergent matrix. From Eq. (35), it is thus immediate that
( ) 0sx k → and ( ) 0se k → as k → ∞ for all (0)sx and (0).e
5. NUMERICAL EXAMPLE
In this section, we consider a fourth-order linear discrete-time control system described by
( 1) ( ) ( )
( ) ( )
x k A x k B u k
y k Cx k
+ = +
=
(37)
where
2.0 0.6 0.2 0.3 1
0.4 0.4 0.9 0.5 1
,
0.1 0.3 0.5 0.1 1
0.7 0.9 0.8 0.8 1
A B
   
   
   = =
   
   
   
and [ ]1 2 1 1 .C = (38)
The eigenvalues of the matrix A are
1 2 32.4964, 1.0994, 0.3203λ λ λ= = = and 4 0.2161.λ = − (39)
From (39), we note that 1 2,λ λ are unstable (slow) eigenvalues and 3 4,λ λ are stable (fast)
eigenvalues of the system matrix .A
For this linear system, the dominant and non-dominant states are calculated. A simple calculation
using the procedure in [9] shows that the first two states{ }1 2,x x are the dominant (slow) states,
while the last two states { }3 4,x x are the non-dominant (fast) states for the given system (37).
Using the procedure described in Section 2, the reduced-order linear model for the given linear
system (37) can be obtained as

10. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.1, No.5, December 2011
94
( 1) ( ) ( )
( ) ( ) ( )
s s s
s s
x k A x k B u k
y k C x k D u k
∗ ∗
∗ ∗
+ = +
= +
(40)
where
2.0077 1.1534 0.8352
,
0.3848 1.5581 1.1653
s sA B∗ ∗   
= =   
   
[ ]1.0092 4.0011sC∗
= and 0.2904sD∗
= −
The step responses of the original plant and the reduced order plant are plotted in Figure 1, which
validates the reduced-order model obtained for the given linear plant.
Figure 1. Step Responses for the Original and Reduced Order Linear Systems
We note also that the reduced-order linear system (40) is completely controllable and completely
observable. Thus, reduced-order observers and observer-based controllers for this plant can be
easily constructed as detailed in Sections 3 and 4.
6. CONCLUSIONS
In this paper, using the dominant state analysis of the given large-scale linear plant, we obtained
the reduced-order model of the linear plant. Then we derived sufficient conditions for the design
of observer-based reduced-order controllers. The observer-based reduced order controllers are
constructed by combining the reduced order controllers for the original linear system which

11. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.1, No.5, December 2011
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require the dominant state of the original system and reduced order observers for the original
linear system which provide an exponential estimate of the dominant state of the original linear
system. We also established a separation principle in this paper which shows that the pole
placement problem and observer problem are independent of each other.
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Authors
Dr. V. Sundarapandian obtained his Doctor of Science degree in Electrical and
Systems Engineering from Washington University, Saint Louis, USA under the guidance
of Late Dr. Christopher I. Byrnes (Dean, School of Engineering and Applied Science) in
1996. He is currently Professor in the Research and Development Centre at Vel Tech Dr.
RR & Dr. SR Technical University, Chennai, Tamil Nadu, India. He has published over
210 refereed international publications. He has published over 100 papers in National
Conferences and over 50 papers in International Conferences. He is the Editor-in-Chief
of International Journal of Mathematics and Scientific Computing, International Journal
of Instrumentation and Control Systems, International Journal of Control Systems and Computer
Modelling, etc. His research interests are Linear and Nonlinear Control Systems, Chaos Theory and
Control, Soft Computing, Optimal Control, Process Control, Operations Research, Mathematical
Modelling, Scientific Computing using MATLAB etc.
Mrs. M. Kavitha received the B.Sc. degree in Mathematics from Bharatidasan
University, Tiruchirappalli, Tamil Nadu in 1998, M.Sc. degree in Mathematics from
Bharatidasan University, Tiruchirappalli, Tamil Nadu in 2000, P.G. Diploma in
Computer Applications from Bharatidasan University, Tiruchirappalli, Tamil Nadu,
M.Phil in Mathematics from Madurai Kamaraj University, Madurai, Tamil Nadu in
2004 and currently pursuing PhD degree in Mathematics from Vel Tech Dr. RR & Dr.
SR Technical University, Chennai, India. She has 9 years of teaching experience and
her thrust areas of research are Difference and Differential Equations and Linear
Control Systems.